Theorem imaeq2 4949

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 4886	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4840	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4624	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4624	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2228	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1348   ran crn 4612   |` cres 4613   "cima 4614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by:  imaeq2i  4951  imaeq2d  4953  ssimaex  5557  ssimaexg  5558  isoselem  5799  f1opw2  6055  fopwdom  6814  ssenen  6829  fiintim  6906  fidcenumlemrk  6931  fidcenumlemr  6932  sbthlem2  6935  isbth  6944  ennnfonelemp1  12361  ennnfonelemnn0  12377  ctinfomlemom  12382  ctinfom  12383  tgcn  13002  iscnp4  13012  cnpnei  13013  cnima  13014  cnconst2  13027  cnrest2  13030  cnptoprest  13033  txcnp  13065  txcnmpt  13067  metcnp3  13305